Improved algorithms for splitting full matrix algebras

Let $\K$ be an algebraic number field of degree $d$ and discriminant $Δ$ over $\Q$. Let $\A$ be an associative algebra over $\K$ given by structure constants such that $\A\cong M_n(\K)$ holds for some positive integer $n$. Suppose that $d$, $n$ and $|Δ|$ are bounded. In a previous paper a polynomial time ff-algorithm was given to construct explicitly an isomorphism $\A \rightarrow M_n(\K)$. Here we simplify and improve this algorithm in the cases $n\leq 43$, $\K=\Q$, and $n=2$, with $\K=\Q(\sqrt{-1})$ or $\K=\Q(\sqrt{-3})$. The improvements are based on work by Y. Kitaoka and R. Coulangeon on tensor products of lattices.

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